# Properties

 Label 5445.2.a.r Level $5445$ Weight $2$ Character orbit 5445.a Self dual yes Analytic conductor $43.479$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 605) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} - q^{5} - 2 \beta q^{7} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 - q^5 - 2*b * q^7 - b * q^8 $$q + \beta q^{2} + q^{4} - q^{5} - 2 \beta q^{7} - \beta q^{8} - \beta q^{10} - 6 q^{14} - 5 q^{16} - 4 \beta q^{17} + 4 \beta q^{19} - q^{20} - 6 q^{23} + q^{25} - 2 \beta q^{28} + 4 q^{31} - 3 \beta q^{32} - 12 q^{34} + 2 \beta q^{35} + 10 q^{37} + 12 q^{38} + \beta q^{40} + 4 \beta q^{41} + 2 \beta q^{43} - 6 \beta q^{46} + 6 q^{47} + 5 q^{49} + \beta q^{50} + 6 q^{53} + 6 q^{56} - 4 \beta q^{61} + 4 \beta q^{62} + q^{64} + 10 q^{67} - 4 \beta q^{68} + 6 q^{70} + 4 \beta q^{73} + 10 \beta q^{74} + 4 \beta q^{76} + 4 \beta q^{79} + 5 q^{80} + 12 q^{82} - 10 \beta q^{83} + 4 \beta q^{85} + 6 q^{86} + 6 q^{89} - 6 q^{92} + 6 \beta q^{94} - 4 \beta q^{95} - 10 q^{97} + 5 \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 - q^5 - 2*b * q^7 - b * q^8 - b * q^10 - 6 * q^14 - 5 * q^16 - 4*b * q^17 + 4*b * q^19 - q^20 - 6 * q^23 + q^25 - 2*b * q^28 + 4 * q^31 - 3*b * q^32 - 12 * q^34 + 2*b * q^35 + 10 * q^37 + 12 * q^38 + b * q^40 + 4*b * q^41 + 2*b * q^43 - 6*b * q^46 + 6 * q^47 + 5 * q^49 + b * q^50 + 6 * q^53 + 6 * q^56 - 4*b * q^61 + 4*b * q^62 + q^64 + 10 * q^67 - 4*b * q^68 + 6 * q^70 + 4*b * q^73 + 10*b * q^74 + 4*b * q^76 + 4*b * q^79 + 5 * q^80 + 12 * q^82 - 10*b * q^83 + 4*b * q^85 + 6 * q^86 + 6 * q^89 - 6 * q^92 + 6*b * q^94 - 4*b * q^95 - 10 * q^97 + 5*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^5 $$2 q + 2 q^{4} - 2 q^{5} - 12 q^{14} - 10 q^{16} - 2 q^{20} - 12 q^{23} + 2 q^{25} + 8 q^{31} - 24 q^{34} + 20 q^{37} + 24 q^{38} + 12 q^{47} + 10 q^{49} + 12 q^{53} + 12 q^{56} + 2 q^{64} + 20 q^{67} + 12 q^{70} + 10 q^{80} + 24 q^{82} + 12 q^{86} + 12 q^{89} - 12 q^{92} - 20 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^5 - 12 * q^14 - 10 * q^16 - 2 * q^20 - 12 * q^23 + 2 * q^25 + 8 * q^31 - 24 * q^34 + 20 * q^37 + 24 * q^38 + 12 * q^47 + 10 * q^49 + 12 * q^53 + 12 * q^56 + 2 * q^64 + 20 * q^67 + 12 * q^70 + 10 * q^80 + 24 * q^82 + 12 * q^86 + 12 * q^89 - 12 * q^92 - 20 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 0 1.00000 −1.00000 0 3.46410 1.73205 0 1.73205
1.2 1.73205 0 1.00000 −1.00000 0 −3.46410 −1.73205 0 −1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5445.2.a.r 2
3.b odd 2 1 605.2.a.f 2
11.b odd 2 1 inner 5445.2.a.r 2
12.b even 2 1 9680.2.a.bg 2
15.d odd 2 1 3025.2.a.j 2
33.d even 2 1 605.2.a.f 2
33.f even 10 4 605.2.g.h 8
33.h odd 10 4 605.2.g.h 8
132.d odd 2 1 9680.2.a.bg 2
165.d even 2 1 3025.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.f 2 3.b odd 2 1
605.2.a.f 2 33.d even 2 1
605.2.g.h 8 33.f even 10 4
605.2.g.h 8 33.h odd 10 4
3025.2.a.j 2 15.d odd 2 1
3025.2.a.j 2 165.d even 2 1
5445.2.a.r 2 1.a even 1 1 trivial
5445.2.a.r 2 11.b odd 2 1 inner
9680.2.a.bg 2 12.b even 2 1
9680.2.a.bg 2 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5445))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{23} + 6$$ T23 + 6 $$T_{53} - 6$$ T53 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 48$$
$19$ $$T^{2} - 48$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} - 48$$
$43$ $$T^{2} - 12$$
$47$ $$(T - 6)^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 48$$
$67$ $$(T - 10)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2} - 48$$
$83$ $$T^{2} - 300$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 10)^{2}$$